Derivative of Functions

Differentiation and the derivative

Differentiation is the action of computing a derivative. The derivative of a function f(x) of a variable x is a measure of the rate at which the value of the function changes with respect to the change of the variable. It is called the derivative of f with respect to x. If x andy are real numbers, and if the graph of fis plotted against x, the derivative is the slope of this graph at each point.

The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y divided by x is a line. In this case, y=f(x)=mx+b, for real numbersm and b, and the slope m is given by

m=

change in y

change in x

=

Δy

Δx

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Derivatives of elementary functions

Function y=f(x)	Derivative f′(x) off(x)
f(x)=c	f(x)′=0
f(x)=ax+b	f(x)′=a
f(x)=ax2+bx+c	f(x)′=2ax+b
f(x)=xa	f(x)′=a*xa−1
f(x)=√x	f(x)′= 1 2√x
f(x)= a x	f(x)′= −a x2

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Exponential, logarithmic, trigonometric

Function y=f(x)	Derivative f′(x) off(x)
f(x)=ex	f(x)′=ex
f(x)=ln(x)	f(x)′= 1 x
f(x)=loga(x)	f(x)′= 1 xln(a)
f(x)=sinx	f(x)′=cosx
f(x)=cosx	f(x)′=−sinx
f(x)=tgx	f(x)′= 1 cos2x
f(x)=ctgx	f(x)′=− 1 sin2x

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